Properties of the ellipse $$$\frac{x^{2}}{4} + \frac{y^{2}}{5} = 1$$$

Find the center, foci, vertices, co-vertices, major axis length, semi-major axis length, minor axis length, semi-minor axis length, area, circumference, latera recta, length of the latera recta (focal width), focal parameter, eccentricity, linear eccentricity (focal distance), directrices, x-intercepts, y-intercepts, domain, and range of the ellipse $$$\frac{x^{2}}{4} + \frac{y^{2}}{5} = 1$$$.

The equation of an ellipse is $$$\frac{\left(x - h\right)^{2}}{a^{2}} + \frac{\left(y - k\right)^{2}}{b^{2}} = 1$$$, where $$$\left(h, k\right)$$$ is the center, $$$b$$$ and $$$a$$$ are the lengths of the semi-major and the semi-minor axes.

Our ellipse in this form is $$$\frac{\left(x - 0\right)^{2}}{4} + \frac{\left(y - 0\right)^{2}}{5} = 1$$$.

Thus, $$$h = 0$$$, $$$k = 0$$$, $$$a = 2$$$, $$$b = \sqrt{5}$$$.

The standard form is $$$\frac{x^{2}}{2^{2}} + \frac{y^{2}}{\left(\sqrt{5}\right)^{2}} = 1$$$.

The vertex form is $$$\frac{x^{2}}{4} + \frac{y^{2}}{5} = 1$$$.

The general form is $$$5 x^{2} + 4 y^{2} - 20 = 0$$$.

The linear eccentricity (focal distance) is $$$c = \sqrt{b^{2} - a^{2}} = 1$$$.

The eccentricity is $$$e = \frac{c}{b} = \frac{\sqrt{5}}{5}$$$.

The first focus is $$$\left(h, k - c\right) = \left(0, -1\right)$$$.

The second focus is $$$\left(h, k + c\right) = \left(0, 1\right)$$$.

The first vertex is $$$\left(h, k - b\right) = \left(0, - \sqrt{5}\right)$$$.

The second vertex is $$$\left(h, k + b\right) = \left(0, \sqrt{5}\right)$$$.

The first co-vertex is $$$\left(h - a, k\right) = \left(-2, 0\right)$$$.

The second co-vertex is $$$\left(h + a, k\right) = \left(2, 0\right)$$$.

The length of the major axis is $$$2 b = 2 \sqrt{5}$$$.

The length of the minor axis is $$$2 a = 4$$$.

The area is $$$\pi a b = 2 \sqrt{5} \pi$$$.

The circumference is $$$4 b E\left(\frac{\pi}{2}\middle| e^{2}\right) = 4 \sqrt{5} E\left(\frac{1}{5}\right)$$$.

The focal parameter is the distance between the focus and the directrix: $$$\frac{a^{2}}{c} = 4$$$.

The latera recta are the lines parallel to the minor axis that pass through the foci.

The first latus rectum is $$$y = -1$$$.

The second latus rectum is $$$y = 1$$$.

The endpoints of the first latus rectum can be found by solving the system $$$\begin{cases} 5 x^{2} + 4 y^{2} - 20 = 0 \\ y = -1 \end{cases}$$$ (for steps, see system of equations calculator).

The endpoints of the first latus rectum are $$$\left(- \frac{4 \sqrt{5}}{5}, -1\right)$$$, $$$\left(\frac{4 \sqrt{5}}{5}, -1\right)$$$.

The endpoints of the second latus rectum can be found by solving the system $$$\begin{cases} 5 x^{2} + 4 y^{2} - 20 = 0 \\ y = 1 \end{cases}$$$ (for steps, see system of equations calculator).

The endpoints of the second latus rectum are $$$\left(- \frac{4 \sqrt{5}}{5}, 1\right)$$$, $$$\left(\frac{4 \sqrt{5}}{5}, 1\right)$$$.

The length of the latera recta (focal width) is $$$\frac{2 a^{2}}{b} = \frac{8 \sqrt{5}}{5}$$$.

The first directrix is $$$y = k - \frac{b^{2}}{c} = -5$$$.

The second directrix is $$$y = k + \frac{b^{2}}{c} = 5$$$.

The x-intercepts can be found by setting $$$y = 0$$$ in the equation and solving for $$$x$$$ (for steps, see intercepts calculator).

x-intercepts: $$$\left(-2, 0\right)$$$, $$$\left(2, 0\right)$$$

The y-intercepts can be found by setting $$$x = 0$$$ in the equation and solving for $$$y$$$: (for steps, see intercepts calculator).

y-intercepts: $$$\left(0, - \sqrt{5}\right)$$$, $$$\left(0, \sqrt{5}\right)$$$

The domain is $$$\left[h - a, h + a\right] = \left[-2, 2\right]$$$.

The range is $$$\left[k - b, k + b\right] = \left[- \sqrt{5}, \sqrt{5}\right]$$$.

Standard form/equation: $$$\frac{x^{2}}{2^{2}} + \frac{y^{2}}{\left(\sqrt{5}\right)^{2}} = 1$$$A.

Vertex form/equation: $$$\frac{x^{2}}{4} + \frac{y^{2}}{5} = 1$$$A.

General form/equation: $$$5 x^{2} + 4 y^{2} - 20 = 0$$$A.

First focus-directrix form/equation: $$$x^{2} + \left(y + 1\right)^{2} = \frac{\left(y + 5\right)^{2}}{5}$$$A.

Second focus-directrix form/equation: $$$x^{2} + \left(y - 1\right)^{2} = \frac{\left(y - 5\right)^{2}}{5}$$$A.

Graph: see the graphing calculator.

Center: $$$\left(0, 0\right)$$$A.

First focus: $$$\left(0, -1\right)$$$A.

Second focus: $$$\left(0, 1\right)$$$A.

First vertex: $$$\left(0, - \sqrt{5}\right)\approx \left(0, -2.23606797749979\right)$$$A.

Second vertex: $$$\left(0, \sqrt{5}\right)\approx \left(0, 2.23606797749979\right)$$$A.

First co-vertex: $$$\left(-2, 0\right)$$$A.

Second co-vertex: $$$\left(2, 0\right)$$$A.

Major axis length: $$$2 \sqrt{5}\approx 4.472135954999579$$$A.

Semi-major axis length: $$$\sqrt{5}\approx 2.23606797749979$$$A.

Minor axis length: $$$4$$$A.

Semi-minor axis length: $$$2$$$A.

Area: $$$2 \sqrt{5} \pi\approx 14.049629462081453$$$A.

Circumference: $$$4 \sqrt{5} E\left(\frac{1}{5}\right)\approx 13.318334443130703$$$A.

First latus rectum: $$$y = -1$$$A.

Second latus rectum: $$$y = 1$$$A.

Endpoints of the first latus rectum: $$$\left(- \frac{4 \sqrt{5}}{5}, -1\right)\approx \left(-1.788854381999832, -1\right)$$$, $$$\left(\frac{4 \sqrt{5}}{5}, -1\right)\approx \left(1.788854381999832, -1\right)$$$A.

Endpoints of the second latus rectum: $$$\left(- \frac{4 \sqrt{5}}{5}, 1\right)\approx \left(-1.788854381999832, 1\right)$$$, $$$\left(\frac{4 \sqrt{5}}{5}, 1\right)\approx \left(1.788854381999832, 1\right)$$$A.

Length of the latera recta (focal width): $$$\frac{8 \sqrt{5}}{5}\approx 3.577708763999664$$$A.

Focal parameter: $$$4$$$A.

Eccentricity: $$$\frac{\sqrt{5}}{5}\approx 0.447213595499958$$$A.

Linear eccentricity (focal distance): $$$1$$$A.

First directrix: $$$y = -5$$$A.

Second directrix: $$$y = 5$$$A.

x-intercepts: $$$\left(-2, 0\right)$$$, $$$\left(2, 0\right)$$$A.

y-intercepts: $$$\left(0, - \sqrt{5}\right)\approx \left(0, -2.23606797749979\right)$$$, $$$\left(0, \sqrt{5}\right)\approx \left(0, 2.23606797749979\right)$$$A.

Domain: $$$\left[-2, 2\right]$$$A.

Range: $$$\left[- \sqrt{5}, \sqrt{5}\right]\approx \left[-2.23606797749979, 2.23606797749979\right]$$$A.